With BYU (rated #1 most "conservative" law school in the nation), while its not considered "Ivy League", it IS nevertheless considered a "1st tier" law school in the United States (just not one of the elitist ones that are super duper selective), and a degree from BYU is looked upon highly favorably. Lots of BYU graduates have gone to become major public figures, so it would be hard to argue someone with a law degree from BYU lacks the gravitas to be a federal judge. The only downside is the school is pretty much known as exclusively Mormon, even though they certainly have non-Mormon students and alumni from time to time.
The second most "conservative" law school in the country, if you look at the rankings, is Ave Maria School of Law. It is a small, very conservative, traditionalist Roman Catholic college in Naples, FL. I think pretty much all their lawyers end up representing Catholic church clients. They are ranked in the "4th tier" of Law Schools by US News & World Report, and would be considered a non-starter for a prestigious career on our nation's highest court.
George W. Bush tried a non-Ivy Leaguer with Harriet Miers (law degree from Southern Methodist University), which backfired, big time.
Amy Coney Barrett is from the "NOT Ivy League, but nevertheless considered 1st tier and prestigious" category by having a law degree from Notre Dame. While "Catholic" on paper, that university (honorary degree for Obama!) is certainly far from conservative. I think she'd probably be more conservative than Kavanaugh or Gorsuch, but the Dems could very easily embark on an endless smear camapign to create a caricature of her to voters like they did to Christine O'Donnell and Sarah Palin, painting her as some kind of kooky religious zealot, and if she was appointed for Ginsburg's seat, it would make the Kavanaugh hearings look like a summer breeze in comparison.
I think U.S. Grant is the last President to appoint a non-Ivy Leaguer as Chief Justice, and although he "succeeded" in getting the nominee confirmed, he appointed some obscure lawyer friend from Ohio who turned out to be a mediocre CJ, and people agree that guy was the wrong choice for the role.
With Nixon, the rejection of G. Harrold Carswell wasn't seen so much that he wasn't an Ivy League, but that he was a fundamentalist from the deep south... the same reason they soured at Nixon's first pick, Ivy-Leaguer Clement Haynsworth. Nixon accused the Senate of having a bias against judges from the deep south. We then ended up with turncoat Harry Blackmun on SCOTUS. Like Sandra Day O'Connor and Neil Gorsuch, he voted with the conservative bloc "90% of the time" his first few years on the court and was touted as a big "accomplishment" for Nixon's goal of a "strict constructionist" court, then showed his true colors and went left once a bunch of "landmark" social liberal cases went before the court like Roe v. Wade.
I can excuse Nixon's failure with Blackmun, and Reagan's failure with Kennedy, moreso than the Gorsuch failure, because at least they TRIED at first to get a much more clearly conservative judge, only to see the Senate kill off TWO of their nominees in a row, and then settled for a "meh" judge instead, as their THIRD choice for the job. But Gorsuch was hardly Trump's third choice for the job.
I disagree. People who remain conservative after three years at an elite law school are less likely to “evolve” later on than those who did not have to withstand daily challenges to their worldview. And had Samuel Alito or Clarence Thomas attended Liberty University Law School instead of Yale Law School they never would have been confirmed to circuit court, much less to SCOTUS.
Also, to the extent that there is a correlation between intellectual and academic ability and the law schools at which applicants are accepted (and there is, albeit obviously not a perfect one), it is preferable to appoint conservatives who graduated from more selective law schools than those from less selective law schools, since the ability to make clever arguments and write convincingly like a Scalia could do is just as important as is voting the right way. Scalia was the sole dissenter in Morrison v. Olsen, and his dissent arguing that the Independent Counsel law violated separation of powers (given that Article II of the U.S. Constitution provides that the executive power is vested in the president) convinced so many people over the next couple of decades (I can tell you that even liberal students at Yale Law School had to admit that he had made the better argument) that Congress eventuallly let the statute lapse due to constitutional concerns.
We need judges who are smart (and acknowledged to be so), conservative and confirmable. The easiest way to accomplish that is to nominate conservatives who graduated from elite law schools and have been vetted by the Federalist Society, which is pretty much the way that President Trump and Senate Majority Leader McConnell have been able to remake the federal judiciary.
Of the Ivy Leaguers, someone actually looked at ALL the SCOTUS judges AND their law clerks over the years, where they went on the political spectrum, and what law school they graduated from, to rank the Ivy League school with the most "liberal" graudates and the Ivy League school with the most "conservative" graduates.
The results really aren't surprising though. Berkley produced the most far-left people on SCOTUS, and the University of Virginia produced the most conservative:
I really don't think see Ivy League by itself being an issue though. Rehnquist & Scalia were Harvard men, Alito and Thomas are Yale.
If you cross tabulate "1st tier law school" with "top 25 most conservative law schools" in the U.S., you get some slim pickings. I think I found 4 that overlap. The options are as follows:
Brigham Young University (J. Reuben Clark Law School)
University of Virginia Law School
University of Alabama Law School
George Mason University-Antonin Scalia Law School